\newcommand{\Epsilonvec}{{\bf{E}}}
\newcommand{\eyemat}{I}

\newcommand{\natbasis}{\mathbf{e}}

\def\bi{\begin{itemize}}
\def\ei{\end{itemize}}
\def\benum{\begin{enumerate}}
\def\eenum{\end{enumerate}}
\def\biplus{\bi[<+->]}
\def\benumplus{\benum[<+->]}
\def\be{\begin{equation*}}
\def\ee{\end{equation*}}
\def\highlight{\magenta}
\def\bad{\red}
\def\conv{\mathrm{conv}}
\def\E{\mathbb{E}}
\def\mF{\mathcal{F}}
\def\mC{\mathcal{C}}
\def\mD{\mathcal{D}}
\def\mP{\mathcal{P}}
\def\mX{\mathcal{X}}
\def\mB{\mathcal{B}}
\def\mG{\mathcal{G}}
\def\mV{\mathcal{V}}
\def\mQ{\mathcal{Q}}
\def\mT{\mathcal{T}}
\def\mW{\mathcal{W}}
\def\l{\left}
\def\r{\right}
\def\hx{\widehat{x}}
\def\xh{\widehat{\mathbf{x}}}
\def\x{\mathbf{x}}
\def\s{\mathbf{s}}
\def\mXh{\widehat{\mathcal{X}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newcommand{\thetamin}{\ensuremath{\theta^*_{min}}}
%\newcommand{\kdim}{\ensuremath{k}}
%\newcommand{\minval}{\ensuremath{\beta}_{min}}
%\newcommand{\regpar}{\ensuremath{\rho}}
%\newcommand{\spgam}{\ensuremath{\gamma}}
\newcommand{\Rmat}{\ensuremath{{R}}}
\newcommand{\Rmatexp}{\ensuremath{\text{Poly}(R,2)}}
\newcommand{\Y}{\ensuremath{Y}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\betadummy}{\ensuremath{\hat{\beta}^{HCA}}}
\newcommand{\betahcadummy}{\ensuremath{\hat{\beta}^{HCA}_{hca}}}
\newcommand{\betaols}{\ensuremath{\hat{\beta}^{APM}}}
\newcommand{\betahcaols}{\ensuremath{\hat{\beta}^{APM}_{hca}}}
\newcommand{\betahca}{\ensuremath{{\beta^*}_{hca}}}

\newcommand{\betalamhca}{\ensuremath{\hat{\beta}^{\lambdavec}_{hca}}}
\newcommand{\betalam}{\ensuremath{\hat{\beta}^{\lambdavec}}}
\newcommand{\zzerolam}{\ensuremath{\hat{\beta}^{\lambdavec}_0}}
\newcommand{\zlam}{\ensuremath{\hat{\beta}^{\lambdavec}}}
\newcommand{\zlamrescaled}{\ensuremath{\hat{\beta}^{\lambdavec, \text{rescaled}}}}
\newcommand{\betalammedian}{\ensuremath{{\hat{\beta}^{\lambdavec}_{median}}}}

\newcommand{\pdimstar}{\ensuremath{{\tilde{\pdim}}}}
\newcommand{\zstar}{\ensuremath{{z^{*}}}}


\newcommand{\betalamhcaapx}{\ensuremath{\beta^{\sigma}_{hca}}}
\newcommand{\betalamapx}{\ensuremath{\beta^{\sigma}}}
\newcommand{\zzerolamapx}{\ensuremath{z^{\sigma}_0}}
\newcommand{\zlamapx}{\ensuremath{z^\sigma}}
\newcommand{\betalammedianapx}{\ensuremath{\beta^{\sigma}_{median}}}

\newcommand{\shrink}{\textrm{Shr}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\thetamin}{\ensuremath{\theta^*_{min}}}
\newcommand{\mutinc}{\ensuremath{\alpha}}
\newcommand{\pdim}{\ensuremath{p}}
\newcommand{\ddim}{\ensuremath{{d}}}
\newcommand{\tsize}{\ensuremath{T}}
\newcommand{\numobs}{\ensuremath{n}}
\newcommand{\kdim}{\ensuremath{k}}
\newcommand{\minval}{\ensuremath{\beta}_{min}}
\newcommand{\regpar}{\ensuremath{\rho}}
\newcommand{\regparn}{\ensuremath{\regpar_\numobs}}
\newcommand{\spgam}{\ensuremath{\gamma}}
\newcommand{\betastar}{\ensuremath{{\beta^*}}}
\newcommand{\Xmat}{\ensuremath{X}}
\newcommand{\Sset}{\ensuremath{S}}
\newcommand{\Ssetcomp}{\ensuremath{S^{\complement}}}
\newcommand{\betaexpanded}[1]{\ensuremath{\beta_{#1}(\prob_{\numobs}^\Sset)}}
\newcommand{\probempiricalcomp}{\ensuremath{\prob_\numobs^{{\Ssetcomp}}}}
\newcommand{\toracle}{\ensuremath{\tilde{t}}}
\newcommand{\that}{\ensuremath{\hat{t}}}
\newcommand{\betahat}{\ensuremath{\hat{\beta}}}
\newcommand{\zhat}{\ensuremath{\hat{z}}}
\newcommand{\lambdapm}{\ensuremath{\vec{\lambda}PM}}
\newcommand{\lambdavec}{\ensuremath{\vec{\lambda}}}

\newcommand{\betahcastar}{\ensuremath{\beta^{*}_{hca}}}

\newcommand{\absoluteerr}{\ensuremath{|\hat{E_i}|}}
